these simulation numerique et modelisation de l'ecoulement autour ...

2.1.1. Injection through the hole
LES of a bi-periodic turbulent flow with effusion 7
2.1. Generation of a periodic flow with effusion
In experiments, channels are bounded by impermeable walls at the top and at the bottom.
If used in conjunction with periodic boundary conditions in the tangential directions,
this outer condition prevents the flow from reaching a statistically steady state with effusion,
because the net mass flux through the perforation tends to eliminate the pressure
drop between the cold and the hot domains. In the present simulations, characteristicbased
freestream boundary conditions (Thompson 1990) are used at the top and bottom
ends of the domain in order to impose the appropriate mean vertical flow rate. Note that
Mendez, Nicoud & Miron (2005) compared this boundary condition-based strategy to
another, source terms-based, methodology and that the independence of the results on
the method used to sustain the effusion was demonstrated.
2.1.2. Primary and secondary flows
For classical, periodic channel or pipe flows simulations, a volumetric source term
S (ρ U) is added to the streamwise momentum conservation equation in order to mimic
the effect of the mean streamwise pressure gradient that would exist in a non-periodic
configuration. The source term is constant over space. For example, it can have the
following form:
S (ρ U) = (ρ U target − ρ U mean )
(2.1)
τ
The source term compares a target momentum value, ρ U target , with the spatial-averaged
momentum in the channel, ρ U mean . The time scale τ characterises the relaxation of
ρ U mean towards its target value. This approach can be generalised to the case of an
effusion configuration, making use of a source term of the previous form in each channel
to generate the primary and the secondary flows and no source term within the hole. Note
however that no source term is required in the aspiration side since the target velocity
for the secondary flow can be imposed through the boundary condition at the bottom
end of the domain; it is then convected throughout the bottom channel without the need
of extra external forcing.
2.2. Numerical simulations
The smallest domain that can reproduce the geometry of an infinite plate with staggered
perforations contains only one hole and is diamond-shaped (see figure 2). In the present
isothermal simulations, the computational domain is divided into two channels. The
upper one, denoted by ‘1’, represents the combustion chamber, with a primary flow of
’hot’ gases. The second one, denoted by ‘2’, represents the casing, with a secondary flow
of ’cooling’ air. The height of the channels are h 1 = 24 d and h 2 = 10 d respectively,
where d = 5 mm is the diameter of the cylindrical aperture. The upper and lower limits
of the domain are far enough from the zone of interest to avoid any spurious effect of
boundary conditions on the flow near the perforated plate. The channels are separated
by a perforated plate of thickness 10 mm, the aperture being angled at α g = 30 ◦ with
the plate, in the streamwise direction, without any spanwise orientation. The thickness
of the plate being 10 mm and holes being angled at 30 ◦ with the plate, the hole lengthto-diameter
ratio is 4. The diagonals of the computational domain are z = 0 and x = 0
and their lengths equal the hole-to-hole distance, viz. 11.68 d in the streamwise direction
(z = 0) and 6.74 d in the spanwise direction (x = 0). The centre of the hole is located
at x = 0, y = 0, z = 0 for the injection side (hole outlet) and at x = −3.46 d, y = −2 d,
z = 0 for the suction side (hole inlet).